Largest trapezoid that can be inscribed in a semicircle

Given a semicircle of radius r, the task is to find the largest trapezoid that can be inscribed in the semicircle, with base lying on the diameter.

Examples:

Input: r = 5
Output: 32.476

Input: r = 8
Output: 83.1384

Recommended: Please try your approach on {IDE} first, before moving on to the solution. Approach: Let r be the radius of the semicircle, x be the lower edge of the trapezoid, and y the upper edge, & h be the height of the trapezoid.
Now from the figure,

r^2 = h^2 + (y/2)^2

or, 4r^2 = 4h^2 + y^2

y^2 = 4r^2 – 4h^2

y = 2√(r^2 – h^2)

We know, Area of Trapezoid, A = (x + y)*h/2

So, A = hr + h√(r^2 – h^2)

taking the derivative of this area function with respect to h, (noting that r is a constant since we are given the semicircle of radius r to start with)

dA/dh = r + √(r^2 – h^2) – h^2/√(r^2 – h^2)

To find the critical points we set the derivative equal to zero and solve for h, we get

h = √3/2 * r

So, x = 2 * r & y = r

So, A = (3 * √3 * r^2)/4

Below is the implementation of above approach:

C++

 // C++ Program to find the biggest trapezoid // which can be inscribed within the semicircle #include using namespace std;    // Function to find the area // of the biggest trapezoid float trapezoidarea(float r) {        // the radius cannot be negative     if (r < 0)         return -1;        // area of the trapezoid     float a = (3 * sqrt(3) * pow(r, 2)) / 4;        return a; }    // Driver code int main() {     float r = 5;     cout << trapezoidarea(r) << endl;     return 0; }

Java

 // Java Program to find the biggest trapezoid // which can be inscribed within the semicircle    import java.util.*; import java.lang.*; import java.io.*;    class GFG{ // Function to find the area // of the biggest trapezoid static float trapezoidarea(float r) {        // the radius cannot be negative     if (r < 0)         return -1;        // area of the trapezoid     float a = (3 * (float)Math.sqrt(3)              * (float)Math.pow(r, 2)) / 4;        return a; }    // Driver code public static void main(String args[]) {     float r = 5;     System.out.printf("%.3f",trapezoidarea(r)); } }

Python 3

 # Python 3 Program to find the biggest trapezoid  # which can be inscribed within the semicircle     # from math import everything from math import *    # Function to find the area  # of the biggest trapezoid  def trapezoidarea(r) :        # the radius cannot be negative      if r < 0 :         return -1        # area of the trapezoid     a = (3 * sqrt(3) * pow(r,2)) / 4        return a       # Driver code      if __name__ == "__main__" :        r = 5        print(round(trapezoidarea(r),3))       # This code is contributed by ANKITRAI1

C#

 // C# Program to find the biggest  // trapezoid which can be inscribed  // within the semicircle using System;    class GFG { // Function to find the area // of the biggest trapezoid static float trapezoidarea(float r) {        // the radius cannot be negative     if (r < 0)         return -1;        // area of the trapezoid     float a = (3 * (float)Math.Sqrt(3) *                     (float)Math.Pow(r, 2)) / 4;        return a; }    // Driver code public static void Main() {     float r = 5;     Console.WriteLine("" + trapezoidarea(r)); } }    // This code is contributed  // by inder_verma

PHP



Output:

32.476

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